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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtrfbas 17501* Conditions for the trace of a filter base  F to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  A  C_  Y )  ->  ( ( Ft  A )  e.  ( fBas `  A ) 
 <-> 
 A. v  e.  F  ( v  i^i  A )  =/=  (/) ) )
 
11.2.2  Filters
 
Syntaxcfil 17502 Extend class notation with the set of filters on a set.
 class  Fil
 
Definitiondf-fil 17503* The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in  RR. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |- 
 Fil  =  ( z  e.  _V  |->  { f  e.  ( fBas `  z )  | 
 A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
 
Theoremisfil 17504* The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  <->  ( F  e.  ( fBas `  X )  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
 
Theoremfilfbas 17505 A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  e.  ( fBas `  X ) )
 
Theorem0nelfil 17506 The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  -.  (/)  e.  F )
 
Theoremfileln0 17507 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  A  =/=  (/) )
 
Theoremfilsspw 17508 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  C_  ~P X )
 
Theoremfilelss 17509 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  A  C_  X )
 
Theoremfilss 17510 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) ) 
 ->  B  e.  F )
 
Theoremfilin 17511 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  e.  F )
 
Theoremfiltop 17512 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  X  e.  F )
 
Theoremisfil2 17513* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  <->  ( ( F 
 C_  ~P X  /\  -.  (/) 
 e.  F  /\  X  e.  F )  /\  A. x  e.  ~P  X ( E. y  e.  F  y  C_  x  ->  x  e.  F )  /\  A. x  e.  F  A. y  e.  F  ( x  i^i  y )  e.  F ) )
 
Theoremisfildlem 17514* Lemma for isfild 17515. (Contributed by Mario Carneiro, 1-Dec-2013.)
 |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ].
 ps ) ) )
 
Theoremisfild 17515* Sufficient condition for a set of the form  { x  e.  ~P A  |  ph } to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  [. A  /  x ].
 ps )   &    |-  ( ph  ->  -.  [. (/)  /  x ]. ps )   &    |-  ( ( ph  /\  y  C_  A  /\  z  C_  y )  ->  ( [. z  /  x ].
 ps  ->  [. y  /  x ].
 ps ) )   &    |-  (
 ( ph  /\  y  C_  A  /\  z  C_  A )  ->  ( ( [. y  /  x ]. ps  /\  [. z  /  x ].
 ps )  ->  [. (
 y  i^i  z )  /  x ]. ps )
 )   =>    |-  ( ph  ->  F  e.  ( Fil `  A ) )
 
Theoremfilfi 17516 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( fi `  F )  =  F )
 
Theoremfilinn0 17517 The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  =/=  (/) )
 
Theoremfilintn0 17518 A filter has the finite intersection property. Remark below definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e.  Fin ) )  ->  |^| A  =/=  (/) )
 
Theoremfiln0 17519 The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  =/=  (/) )
 
Theoreminfil 17520 The intersection of two filters is a filter. Use fiint 7101 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  G  e.  ( Fil `  X ) )  ->  ( F  i^i  G )  e.  ( Fil `  X ) )
 
Theoremsnfil 17521 A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
 
Theoremfbasweak 17522 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  F  C_  ~P Y  /\  Y  e.  V ) 
 ->  F  e.  ( fBas `  Y ) )
 
Theoremsnfbas 17523 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
 
Theoremfsubbas 17524 A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( X  e.  V  ->  ( ( fi `  A )  e.  ( fBas `  X )  <->  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi
 `  A ) ) ) )
 
Theoremfbasfip 17525 A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  -.  (/)  e.  ( fi
 `  F ) )
 
Theoremfbunfip 17526* A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  Y ) )  ->  ( -.  (/)  e.  ( fi
 `  ( F  u.  G ) )  <->  A. x  e.  F  A. y  e.  G  ( x  i^i  y )  =/=  (/) ) )
 
Theoremfgval 17527* The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( X filGen F )  =  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
 
Theoremelfg 17528* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( A  e.  ( X filGen F )  <->  ( A  C_  X  /\  E. x  e.  F  x  C_  A ) ) )
 
Theoremssfg 17529 A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  F  C_  ( X filGen F ) )
 
Theoremfgss 17530 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X )  /\  F  C_  G )  ->  ( X filGen F )  C_  ( X filGen G ) )
 
Theoremfgss2 17531* A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X ) )  ->  ( ( X filGen F )  C_  ( X filGen G )  <->  A. x  e.  F  E. y  e.  G  y  C_  x ) )
 
Theoremfgfil 17532 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( X filGen F )  =  F )
 
Theoremelfilss 17533* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  F  <->  E. t  e.  F  t 
 C_  A ) )
 
Theoremfilfinnfr 17534 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  S  e.  F  /\  S  e.  Fin )  ->  |^| F  =/=  (/) )
 
Theoremfgcl 17535 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( X filGen F )  e.  ( Fil `  X ) )
 
Theoremfgabs 17536 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  Y  C_  X )  ->  ( X filGen ( Y
 filGen F ) )  =  ( X filGen F ) )
 
Theoremneifil 17537 The neighborhoods of a non empty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/= 
 (/) )  ->  (
 ( nei `  J ) `  S )  e.  ( Fil `  X ) )
 
Theoremfilunibas 17538 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  U. F  =  X )
 
Theoremfilunirn 17539 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
 
Theoremfilcon 17540 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( F  u.  { (/) } )  e.  Con )
 
Theoremfbasrn 17541* Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  C  =  ran  (  x  e.  B  |->  ( F
 " x ) )   =>    |-  ( ( B  e.  ( fBas `  X )  /\  F : X --> Y  /\  Y  e.  V )  ->  C  e.  ( fBas `  Y ) )
 
Theoremfiluni 17542* The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  C_  ( Fil `  X )  /\  F  =/=  (/)  /\  A. f  e.  F  A. g  e.  F  ( f  u.  g )  e.  F )  ->  U. F  e.  ( Fil `  X ) )
 
Theoremtrfil1 17543 Conditions for the trace of a filter  L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  =  U. ( Lt  A ) )
 
Theoremtrfil2 17544* Conditions for the trace of a filter  L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( Fil `  A ) 
 <-> 
 A. v  e.  L  ( v  i^i  A )  =/=  (/) ) )
 
Theoremtrfil3 17545 Conditions for the trace of a filter  L to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( Fil `  A ) 
 <->  -.  ( Y  \  A )  e.  L ) )
 
Theoremtrfilss 17546 If  A is a member of the filter, then the filter truncated to  A is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  ( Ft  A )  C_  F )
 
Theoremfgtr 17547 If  A is a member of the filter, then truncating  F to  A and regenerating the behavior outside  A using 
filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  ( X filGen ( Ft  A ) )  =  F )
 
Theoremtrfg 17548 The trace operation and the 
filGen operation are inverses to one another in some sense, with  filGen growing the base set and ↾t shrinking it. See fgtr 17547 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  A )  /\  A  C_  X  /\  X  e.  V )  ->  ( ( X filGen F )t  A )  =  F )
 
Theoremtrnei 17549 The trace, over a set  A, of the filter of the neighborhoods of a point  P is a filter iff  P belongs to the closure of  A. (This is trfil2 17544 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  ( P  e.  ( ( cls `  J ) `  A )  <->  ( ( ( nei `  J ) `  { P } )t  A )  e.  ( Fil `  A ) ) )
 
Theoremcfinfil 17550* Relative complements of the finite parts of an infinite set is a filter. When  A  =  NN the set of the relative complements is called Frechet's filter and is used to define the concept of limit of a sequence. (Contributed by FL, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  V  /\  A  C_  X  /\  -.  A  e.  Fin )  ->  { x  e. 
 ~P X  |  ( A  \  x )  e.  Fin }  e.  ( Fil `  X )
 )
 
Theoremcsdfil 17551* The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  dom  card  /\  om  ~<_  X ) 
 ->  { x  e.  ~P X  |  ( X  \  x )  ~<  X }  e.  ( Fil `  X ) )
 
Theoremsupfil 17552* The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  B  C_  x }  e.  ( Fil `  A ) )
 
Theoremzfbas 17553 The set of upper integer sets is a filter base on  ZZ, which corresponds to convergence of sequences on  ZZ. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 ran  ZZ>=  e.  ( fBas `  ZZ )
 
Theoremuzrest 17554 The restriction of the set of upper integers to an upper integer set is the set of upper integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  =  (
 ZZ>= " Z ) )
 
Theoremuzfbas 17555 The set of upper integer sets based at a point in a fixed upper integer set like  NN is a filter base on  NN, which corresponds to convergence of sequences on 
NN. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( ZZ>= " Z )  e.  ( fBas `  Z )
 )
 
11.2.3  Ultrafilters
 
Syntaxcufil 17556 Extend class notation with the ultrafilters-on-a-set function.
 class  UFil
 
Syntaxcufl 17557 Extend class notation with the ultrafilter lemma.
 class UFL
 
Definitiondf-ufil 17558* Define the set of ultrafilters on a set. An ultrafilter is a filter that gives a definite result for every subset. (Contributed by Jeff Hankins, 30-Nov-2009.)
 |- 
 UFil  =  ( g  e.  _V  |->  { f  e.  ( Fil `  g )  | 
 A. x  e.  ~P  g ( x  e.  f  \/  ( g 
 \  x )  e.  f ) } )
 
Definitiondf-ufl 17559* Define the class of base sets for which the ultrafilter lemma filssufil 17569 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |- UFL 
 =  { x  |  A. f  e.  ( Fil `  x ) E. g  e.  ( UFil `  x ) f  C_  g }
 
Theoremisufil 17560* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( F  e.  ( UFil `  X )  <->  ( F  e.  ( Fil `  X )  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X 
 \  x )  e.  F ) ) )
 
Theoremufilfil 17561 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  F  e.  ( Fil `  X ) )
 
Theoremufilss 17562 For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F )
 )
 
Theoremufilb 17563 The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F 
 <->  ( X  \  S )  e.  F )
 )
 
Theoremufilmax 17564 Any filter finer than an ultrafilter is actually equal to it. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  ->  F  =  G )
 
Theoremisufil2 17565* The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  <->  ( F  e.  ( Fil `  X )  /\  A. f  e.  ( Fil `  X ) ( F  C_  f  ->  F  =  f ) ) )
 
Theoremufprim 17566 An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  A  C_  X  /\  B  C_  X )  ->  ( ( A  e.  F  \/  B  e.  F ) 
 <->  ( A  u.  B )  e.  F )
 )
 
Theoremtrufil 17567 Conditions for the trace of an ultrafilter  L to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( L  e.  ( UFil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( UFil `  A ) 
 <->  A  e.  L ) )
 
Theoremfilssufilg 17568* A filter is contained in some ultrafilter. This version of filssufil 17569 contains the choice as a hypothesis (in the assumption that  ~P ~P X is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ~P ~P X  e.  dom  card )  ->  E. f  e.  ( UFil `  X ) F  C_  f )
 
Theoremfilssufil 17569* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 8065.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  E. f  e.  ( UFil `  X ) F 
 C_  f )
 
Theoremisufl 17570* Define the (strong) ultrafilter lemma, parameterized over base sets. A set  X satisfies the ultrafilter lemma if every filter on  X is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( X  e.  V  ->  ( X  e. UFL  <->  A. f  e.  ( Fil `  X ) E. g  e.  ( UFil `  X ) f  C_  g ) )
 
Theoremufli 17571* Property of a set that satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( X  e. UFL  /\  F  e.  ( Fil `  X ) )  ->  E. f  e.  ( UFil `  X ) F 
 C_  f )
 
Theoremnumufl 17572 Consequence of filssufilg 17568: a set whose double powerset is well-orderable satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ~P ~P X  e.  dom  card  ->  X  e. UFL )
 
Theoremfiufl 17573 A finite set satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( X  e.  Fin  ->  X  e. UFL )
 
Theoremacufl 17574 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  (CHOICE 
 -> UFL  =  _V )
 
Theoremssufl 17575 If  Y is a subset of  X and filters extend to ultrafilters in  X, then they still do in  Y. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( X  e. UFL  /\  Y  C_  X )  ->  Y  e. UFL )
 
Theoremufileu 17576* If the ultrafilter containing a given filter is unique, the filter is an ultrafilter. (Contributed by Jeff Hankins, 3-Dec-2009.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( F  e.  ( UFil `  X )  <->  E! f  e.  ( UFil `  X ) F 
 C_  f ) )
 
Theoremfilufint 17577* A filter is equal to the intersection of the ultrafilters containing it. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  |^| { f  e.  ( UFil `  X )  |  F  C_  f }  =  F )
 
Theoremuffix 17578* Lemma for fixufil 17579 and uffixfr 17580. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  V  /\  A  e.  X )  ->  ( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } }
 ) ) )
 
Theoremfixufil 17579* The condition describing a fixed ultrafilter always produces an ultrafilter. (Contributed by Jeff Hankins, 9-Dec-2009.) (Revised by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 29-Jul-2015.)
 |-  ( ( X  e.  V  /\  A  e.  X )  ->  { x  e. 
 ~P X  |  A  e.  x }  e.  ( UFil `  X ) )
 
Theoremuffixfr 17580* An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element  A), and a free ultrafilter is called nonprincipal (having empty intersection). Note that examples of free ultrafilters cannot be defined in ZFC without some form of global choice. (Contributed by Jeff Hankins, 4-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( A  e.  |^| F  <->  F  =  { x  e. 
 ~P X  |  A  e.  x } ) )
 
Theoremuffix2 17581* A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( |^| F  =/=  (/)  <->  E. x  e.  X  F  =  { y  e.  ~P X  |  x  e.  y } ) )
 
Theoremuffixsn 17582 The singleton of the generator of a fixed ultrafilter is in the filter. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  F )
 
Theoremufildom1 17583 An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  |^| F  ~<_  1o )
 
Theoremuffinfix 17584* An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  e.  F  /\  S  e.  Fin )  ->  E. x  e.  X  F  =  { y  e.  ~P X  |  x  e.  y } )
 
Theoremcfinufil 17585* An ultrafilter is free iff it contains the Fréchet filter cfinfil 17550 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( |^| F  =  (/)  <->  { x  e.  ~P X  |  ( X  \  x )  e.  Fin }  C_  F ) )
 
Theoremufinffr 17586* An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)
 |-  ( ( X  e.  B  /\  A  C_  X  /\  om  ~<_  A )  ->  E. f  e.  ( UFil `  X ) ( A  e.  f  /\  |^| f  =  (/) ) )
 
Theoremufilen 17587* Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
 |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X ) A. x  e.  f  x  ~~  X )
 
Theoremufildr 17588 An ultrafilter gives rise to a connected door topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
 |-  J  =  ( F  u.  { (/) } )   =>    |-  ( F  e.  ( UFil `  X )  ->  ( J  u.  ( Clsd `  J ) )  =  ~P X )
 
Theoremfin1aufil 17589 There are no definable free ultrafilters in ZFC. However, there are free ultrafilters in some choice-denying constructions. Here we show that given an amorphous set (a.k.a. a Ia-finite I-infinite set)  X, the set of infinite subsets of 
X is a free ultrafilter on  X. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  F  =  ( ~P X  \  Fin )   =>    |-  ( X  e.  (FinIa  \  Fin )  ->  ( F  e.  ( UFil `  X )  /\  |^| F  =  (/) ) )
 
11.2.4  Filter limits
 
Syntaxcfm 17590 Extend class definition to include the neighborhood filter mapping function.
 class  FilMap
 
Syntaxcflim 17591 Extend class notation with a function returning the limit of a filter.
 class  fLim
 
Syntaxcflf 17592 Extend class definition to include the function for filter-based function limits.
 class  fLimf
 
Syntaxcfcls 17593 Extend class definition to include the cluster point function on filters.
 class  fClus
 
Syntaxcfcf 17594 Extend class definition to include the function for cluster points of a function.
 class  fClusf
 
Definitiondf-fm 17595* Define a function that takes a filter to a neighborhood filter of the range. (Since we now allow filter bases to have support smaller than the base set, the function has to come first to ensure that curryings are sets.) (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 20-Jul-2015.)
 |-  FilMap  =  ( x  e. 
 _V ,  f  e. 
 _V  |->  ( y  e.  ( fBas `  dom  f ) 
 |->  ( x filGen ran  (  t  e.  y  |->  ( f " t ) ) ) ) )
 
Definitiondf-flim 17596* Define a function (indexed by a topology  j) whose value is the limits of a filter  f. (Contributed by Jeff Hankins, 4-Sep-2009.)
 |- 
 fLim  =  ( j  e.  Top ,  f  e. 
 U. ran  Fil  |->  { x  e.  U. j  |  ( ( ( nei `  j
 ) `  { x } )  C_  f  /\  f  C_  ~P U. j
 ) } )
 
Definitiondf-flf 17597* Define a function that gives the limits of a function  f in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.)
 |- 
 fLimf  =  ( x  e.  Top ,  y  e. 
 U. ran  Fil  |->  ( f  e.  ( U. x  ^m  U. y )  |->  ( x  fLim  ( ( U. x  FilMap  f ) `
  y ) ) ) )
 
Definitiondf-fcls 17598* Define a function that takes a filter in a topology to its set of cluster points. (Contributed by Jeff Hankins, 10-Nov-2009.)
 |- 
 fClus  =  ( j  e.  Top ,  f  e. 
 U. ran  Fil  |->  if ( U. j  =  U. f ,  |^|_ x  e.  f  ( ( cls `  j ) `  x ) ,  (/) ) )
 
Definitiondf-fcf 17599* Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.)
 |-  fClusf  =  ( j  e. 
 Top ,  f  e.  U.
 ran  Fil  |->  ( g  e.  ( U. j  ^m  U. f )  |->  ( j 
 fClus  ( ( U. j  FilMap  g ) `  f
 ) ) ) )
 
Theoremfmval 17600* Introduce a function that takes a function from a filtered domain to a set and produces a filter which consists of supersets of images of filter elements. The functions which are dealt with by this function are similar to nets in topology. For example, suppose we have a sequence filtered by the filter generated by its tails under the usual natural number ordering. Then the elements of this filter are precisely the supersets of tails of this sequence. Under this definition, it is not too difficult to see that the limit of a function in the filter sense captures the notion of convergence of a sequence. As a result, the notion of a filter generalizes many ideas associated with sequences, and this function is one way to make that relationship precise in Metamath. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 ->  ( ( X  FilMap  F ) `  B )  =  ( X filGen ran  (  y  e.  B  |->  ( F " y ) ) ) )
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